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In this paper, we propose an integral equation approach for pricing an American-style Parisian up-and-out call option under the Black–Scholes framework. The main difficulty of pricing this option lies in the determination of its optimal exercise price, which is a three-dimensional surface, instead of a two-dimensional (2-D) curve as is the case for a “one-touch” barrier option. In our approach, we first reduce the 3-D pricing problem to a 2-D one by using the “moving window” technique developed by Zhu and Chen (2013, Pricing Parisian and Parasian options analytically. Journal of Economic Dynamics and Control, 37(4): 875-896), then apply the Fourier sine transform to the 2-D problem to obtain two coupled integral equations in terms of two unknown quantities: the option price at the asset barrier and the optimal exercise price. Once the integral equations are solved numerically by using an iterative procedure, the calculation of the option price and the hedging parameters is straightforward from their integral representations. Our approach is validated by a comparison between our results and those of the trusted finite difference method. Numerical results are also provided to show some interesting features of the prices of American-style Parisian up-and-out call options and the behaviour of the associated optimal exercise boundaries.

We derive an analytical solution for the value of Parisian up-and-in calls by using the “moving window” technique for pricing European-style Parisian up-and-out calls. Our pricing formula can be applied to both European-style and American-style Parisian up-and-in calls, due to the fact that with an “in” barrier, the option holder cannot do or decide on anything before the option is activated, and once the option is activated it is just a plain vanilla call, which could be of American style or European style.

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